MATH 151 CALCULUS I MIDTERM 1 2023 Q & A CORRECT

1 Math 151 Practice Midterm 1 2023 Calculus I 2 x x Problem 1: Is it the case that for all rational numbers x with x < 1, 1 > 1? If so prove it, if not characterize those rational x such that 1 > 1. Yes, it is the case that for all rational numbers x with x < 1, 1/x > 1. Let x be a rational number such that x < 1. This means that x can be written in the form x = p/q, where p and q are integers and q > p. Since x < 1, we have p/q < 1, which implies that p < q. Now, we want to show that 1/x > 1. We have: 1/x = 1/(p/q) = q/p. Since p < q, we have q/p > 1, which means that 1/x > 1. Therefore, for all rat ional numbers x with x < 1, 1/x > 1. To characterize those rational numbers x such that 1/x > 1, we can say that x must be a rational number such that 0 < x < 1. This is because for x to be less than 1, it must be in the interval (0, 1), and for 1/x to be greater than 1, x must be in the same interval. So, the set of rational numbers x such that 1/x > 1 is the set of rational numbers in the open interval (0, 1).